Efficient 3D Model to Predict Time History of Structural Dynamics in Cold Rolling Mills

Introduced is a new physics-based 3D mathematical model capable of efficiently predicting time histories of the nonlinear structural dynamics in cold rolling mills used to manufacture metal strip and sheet. The described model allows for prediction of transient strip thickness profiles, contact force distributions, and roll-stack deformations due to dynamic disturbances. Formulation of the new 3D model is achieved through combination of the highly-efficient simplified-mixed finite element method with a Newmark-beta direct time integration approach to solve the system of differential equations that governs motion of the roll-stack. In contrast to prior approaches to predict structural dynamics in cold rolling, the presented method abandons several simplifying assumptions and restrictions, including 1D or 2D linear lumped parameter analyses, vertical symmetry, continuous and constant contact between the rolls and strip, as well as inability to model cluster-type mill configurations and accommodate typical profile/flatness control mechanisms used in industry. Following spatial and temporal convergence studies of the undamped step response, and validation of the damped step response, the new model is demonstrated for a 4-high mill equipped with both work-roll bending and work-roll crown, a 6-high mill with continuously-variable-crown (CVC) intermediate rolls, and finally a complex 20-high cluster mill. Solution times on a single computing processor for the damped 4-high and 20-high case studies are just 0.37 seconds and 3.38 seconds per time step, respectively.